Intbar

Diese Seite wird genutzt, um Darstellungsvarianten von Wiki-TeX zu testen.

Große Symbole

Normale Integrale

Serverseitiges MathMl als SVG

Normales Integral über eine Menge mit displaystyle:

$\displaystyle \int _{A}f(x)\mathrm {d} x$

Normales Integral über ein Intervall mit Limits-Parameter mit displaystyle:

$\displaystyle \int \limits _{0}^{\infty }f(x)dx$

Normales Integral über ein Intervall (ohne Limits-Parameter) mit displaystyle:

$\displaystyle \int _{0}^{\infty }f(x)dx$

Normales Integral über ein Intervall mit NoLimits-Parameter mit displaystyle

$\displaystyle \int \nolimits _{0}^{\infty }f(x)dx$

Normales Integral über ein Intervall (ohne Limits-Parameter) mit textstyle: $\textstyle \int _{0}^{\infty }f(x)dx$

Natives MathMl

Normales Integral über eine Menge mit displaystyle:

\int_{A} f (x) d x

Normales Integral über ein Intervall mit Limits-Parameter mit displaystyle:

\int_{0}^{\infty} f (x) d x

Normales Integral über ein Intervall (ohne Limits-Parameter) mit displaystyle:

\int_{0}^{\infty} f (x) d x

Normales Integral über ein Intervall mit NoLimits-Parameter mit displaystyle

\int_{0}^{\infty} f (x) d x

Normales Integral über ein Intervall (ohne Limits-Parameter) mit textstyle: $\int_{0}^{\infty} f (x) d x$

Mittelwertintegrale

Serverseitiges MathMl als SVG

Integral mit zwei Strichen: $\intBar _{A}f(x)\mathrm {d} x$
Integral mit einem Strich: $\intbar _{A}f(x)\mathrm {d} x$
Integral mit einem Strich in Textstyle: $\textstyle \intbar _{A}f(x)\mathrm {d} x$
Integral mit einem Strich und mit Limits: $\intBar \limits _{A}f(x)\mathrm {d} x$

Natives MathMl

intbar in native mode mit textstyle $⨍ ⨎$
intbar in native mode mit displaystyle $⨍ ⨎$
intbar in native mode mit displaystyle und limits $⨍_{0}^{\infty} ⨎$

Weitere Integrale

Serverseitiges MathMl als SVG

$\oint _{\partial \Sigma }\langle F,\tau \rangle \,\mathrm {d} s=\iint _{\Sigma }\langle \operatorname {rot} \,F,\nu \rangle \,\mathrm {d} S=\iint _{\Sigma }\left({\frac {\partial v_{2}}{\partial x}}-{\frac {\partial v_{1}}{\partial y}}\right)\mathrm {d} x\mathrm {d} y$
$\iiint _{V_{\mathrm {Kart} }}f(x,y,z)\,\mathrm {d} x\mathrm {d} y\mathrm {d} z=\iiint _{V_{\mathrm {Kug} }}{\tilde {f}}(r,\theta ,\varphi )\cdot r^{2}\sin \theta \,\mathrm {d} r\mathrm {d} \theta \mathrm {d} \varphi$

Natives MathMl

$\oint_{\partial Σ} ⟨ F, τ ⟩ d s = \iint_{Σ} ⟨ rot F, ν ⟩ d S = \iint_{Σ} (\frac{\partial v_{2}}{\partial x} - \frac{\partial v_{1}}{\partial y}) d x d y$ phab:T382684
$\underset{V_{K a r t}}{∭} f (x, y, z) d x d y d z = \underset{V_{K u g}}{∭} \tilde{f} (r, θ, φ) \cdot r^{2} \sin θ d r d θ d φ$

Volumen und Kurvenintegrale aus Stix macros

$∰_{A} f (x) d x$
$∯_{A} f (x) d x$
$∳_{A} f (x) d x$
$∳_{A} f (x) d x$
$∲_{A} f (x) d x$
$∲_{A} f (x) d x$

Summen

Oplus-Summe

Serverseitiges MathML

$\mathrm {T} ^{(n)}(V):=\bigoplus _{i=0}^{n}V^{\otimes i}$

Natives MathMl

$T^{(n)} (V) : = ⨁_{i = 0}^{n} V^{\otimes i}$

Sigma-Summenzeichen

Serverseitiges MathML

${\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1$
${\frac {1}{\pi }}\sum \limits _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1$
${\frac {1}{\pi }}\sum \nolimits _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1$
$\textstyle {\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1$

Natives MathMl

$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$ phab:T382669
$\frac{1}{π} \sum_{n = 1}^{\infty} \frac{\sin (2 π n x)}{n} = \frac{1}{2} - x, 0 < x < 1$

Weitere Summen und Produkte

Serverseitiges MathML

$F_{N}(s)=\prod _{p\leq N \atop p\ {\text{Primzahl}}}\sum _{k=0}^{\infty }{\frac {f(p^{k})}{p^{ks}}}$
$F_{N}(s)=\prod \limits _{p\leq N \atop p\ {\text{Primzahl}}}\sum _{k=0}^{\infty }{\frac {f(p^{k})}{p^{ks}}}$
$\sum _{I,J}f_{I,J}\mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}:=\sum _{\stackrel {1\leq i_{1}<\ldots <i_{p}\leq p}{1\leq j_{1}<\ldots <j_{q}\leq q}}^{}f_{i_{1},\ldots i_{p},j_{1},\ldots j_{q}}\mathrm {d} z_{i_{1}}\wedge \cdots \wedge \mathrm {d} z_{i_{p}}\wedge \mathrm {d} {\overline {z}}_{j_{1}}\wedge \cdots \wedge \mathrm {d} {\overline {z}}_{j_{q}}$

Natives MathMl

$F_{N} (s) = \prod_{\binom{p \leq N}{p Primzahl}} \sum_{k = 0}^{\infty} \frac{f (p^{k})}{p^{k s}}$
$F_{N} (s) = \prod_{\binom{p \leq N}{p Primzahl}} \sum_{k = 0}^{\infty} \frac{f (p^{k})}{p^{k s}}$
$\sum_{I, J} f_{I, J} d z_{I} \land d {\overline{z}}_{J} : = \sum_{\overset{1 \leq i_{1} < \dots < i_{p} \leq p}{1 \leq j_{1} < \dots < j_{q} \leq q}}^{} f_{i_{1}, \dots i_{p}, j_{1}, \dots j_{q}} d z_{i_{1}} \land \dots \land d z_{i_{p}} \land d {\overline{z}}_{j_{1}} \land \dots \land d {\overline{z}}_{j_{q}}$

Klammern

Eckige Klammern

Serverseitiges MathML

$\gamma \,x+\ln {\bigl [}\Gamma (x+1){\bigr ]}=\sum _{n=1}^{\infty }{\biggl [}{\frac {x}{n}}-\ln {\biggl (}1+{\frac {x}{n}}{\biggr )}{\biggr ]}$

Natives MathMl

$γ x + \ln [Γ (x + 1)] = \sum_{n = 1}^{\infty} [\frac{x}{n} - \ln (1 + \frac{x}{n})]$

Runde Klammern

Serverseitiges MathML

$\bigvee \limits _{i\in I}{\mathcal {A}}_{i}=\sigma \left(\bigcup \limits _{i\in I}{\mathcal {A}}_{i}\right)$

Natives MathMl

$\underset{i \in I}{⋁} 𝒜_{i} = σ (⋃_{i \in I} 𝒜_{i})$

Runde Biggr-Klammern

Serverseitiges MathML

${\biggl (}_{i}{\frac {1}{n}}{\biggr )}$
$\bigcap _{n\in \mathbb {N} }A_{n}={\biggl (}\bigcup _{n\in \mathbb {N} }A_{n}^{\mathsf {c}}{\biggr )}^{\!\!{\mathsf {c}}}$

Natives MathMl

$(_{i} \frac{1}{n})$ phab:T352946
$⋂_{n \in ℕ} A_{n} = (⋃_{n \in ℕ} A_{n}^{c})^{c}$ phab:T352946

Geschweifte Klammern

Serverseitiges MathMl

$\left\{{\begin{array}{rl}{\frac {1}{2\pi }}\ln {|x|}\ ,&n=2,\\-{\frac {1}{(n-2)\,\omega _{n}}}{\frac {1}{|x|^{n-2}}}\ ,&n>2\\\end{array}}\right.$

Natives MathMl

${\begin{aligned} \frac{1}{2 π} \ln | x |, & n = 2, \\ - \frac{1}{(n - 2) ω_{n}} \frac{1}{| x |^{n - 2}}, & n > 2 \end{aligned}$

Buchstaben

Operatorname

Serverseitiges MathML

$\operatorname {Hom}$

Natives MathMl

$Hom$

Bf-Stil

Serverseitiges MathML

${\bf {Spec}},{\bf {A,{\bf {B,{\bf {C,{\bf {D,{\bf {E,{\bf {F,{\bf {G,{\bf {a,{\bf {1,{\bf {23456789}}}}}}}}}}}}}}}}}}}}$

Natives MathMl

$𝐒 𝐩 𝐞 𝐜, 𝐀, 𝐁, 𝐂, 𝐃, 𝐄, 𝐅, 𝐆, 𝐚, 𝟏, 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗$ ~~phab:T382672~~

Mathbb

Serverseitiges MathML

$\mathbb {N} ,\mathbb {Z} ,\mathbb {Q} ,\mathbb {R} ,\mathbb {R} ,\mathbb {C} ,\mathbb {C} ,\mathbb {H} ,\mathbb {a} ,\mathbb {1}$ phab:T279805

Natives MathMl

$ℕ, ℤ, ℚ, ℝ, ℝ, ℂ, ℂ, ℍ, 𝕒, 𝟙$

Mathcal

Serverseitiges MathML

${\mathcal {ABCDEFGHIJKLM}}{\mathcal {N}}{\mathcal {a}}{\mathcal {1}}$

Natives MathMl

$𝒜 ℬ 𝒞 𝒟 ℰ ℱ 𝒢 ℋ ℐ 𝒥 𝒦 ℒ ℳ 𝒩 𝒶 𝟏$

Sonderzeichen

D’Alembert-Operator

Serverseitiges MathMl

$◻$

Natives MathMl

$◻$ phab:T382681

Funktionen

Determinante mit Potenz und Nolimits

Serverseitiges MathMl

$\det \frac{1}{2}$

Natives MathMl

$\det \frac{1}{2}$ phab:T382716 phab:T378257

Determinante mit Potenz ohne Nolimits

Serverseitiges MathMl

$\det^{\frac{1}{2}}$

Natives MathMl

$\det^{\frac{1}{2}}$

Supremum

Serverseitiges MathMl

$e s s \sup$

Natives MathMl

$e s s \sup$

Matrizen

MathMl

$A = (\begin{matrix} 3 & 2 & 1 \\ 1 & 0 & 2 \end{matrix}) \in ℝ^{2 \times 3}$

Native

$A = (\begin{matrix} 3 & 2 & 1 \\ 1 & 0 & 2 \end{matrix}) \in ℝ^{2 \times 3}$

Fließtext (Satz von Atkinson)

MathMl

Nach dem Satz von Atkinson ist ein Operator $A : X \to Y$ genau dann ein Fredholm-Operator, wenn es Operatoren $B_{1}, B_{2}$ und kompakte Operatoren $K_{1}, K_{2}$ gibt, so dass $A B_{1} = I_{Y} - K_{1}$ und $B_{2} A = I_{X} - K_{2}$ gilt, das heißt wenn $A$ modulo kompakter Operatoren invertierbar ist. Insbesondere ist ein beschränkter Operator $A : X \to X$ genau dann ein Fredholm-Operator, wenn seine Klasse $[A]_{𝒞 (X)}$ in der Calkin-Algebra $ℬ (X) / 𝒞 (X)$ invertierbar ist.

Native

Nach dem Satz von Atkinson ist ein Operator $A : X \to Y$ genau dann ein Fredholm-Operator, wenn es Operatoren $B_{1}, B_{2}$ und kompakte Operatoren $K_{1}, K_{2}$ gibt, so dass $A B_{1} = I_{Y} - K_{1}$ und $B_{2} A = I_{X} - K_{2}$ gilt, das heißt wenn $A$ modulo kompakter Operatoren invertierbar ist. Insbesondere ist ein beschränkter Operator $A : X \to X$ genau dann ein Fredholm-Operator, wenn seine Klasse $[A]_{𝒞 (X)}$ in der Calkin-Algebra $ℬ (X) / 𝒞 (X)$ invertierbar ist.

Algin

MathMl

$\begin{aligned} (\frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}}) \cdot (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ = & \frac{\partial^{2}}{\partial r^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{2}{r} \frac{\partial}{\partial r} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ = & v_{r, r r} {\hat{e}}_{r} + v_{θ, r r} {\hat{e}}_{θ} + v_{φ, r r} {\hat{e}}_{φ} + \frac{2}{r} v_{r, r} {\hat{e}}_{r} + \frac{2}{r} v_{θ, r} {\hat{e}}_{θ} + \frac{2}{r} v_{φ, r} {\hat{e}}_{φ} \\ + \frac{1}{r^{2}} \frac{\partial}{\partial θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial}{\partial φ} (v_{r, φ} {\hat{e}}_{r} + \sin θ v_{r} {\hat{e}}_{φ} + v_{θ, φ} {\hat{e}}_{θ} + \cos θ v_{θ} {\hat{e}}_{φ} + v_{φ, φ} {\hat{e}}_{φ} - \sin θ v_{φ} {\hat{e}}_{r} - \cos θ v_{φ} {\hat{e}}_{θ}) \\ = & v_{r, r r} {\hat{e}}_{r} + v_{θ, r r} {\hat{e}}_{θ} + v_{φ, r r} {\hat{e}}_{φ} + \frac{2}{r} v_{r, r} {\hat{e}}_{r} + \frac{2}{r} v_{θ, r} {\hat{e}}_{θ} + \frac{2}{r} v_{φ, r} {\hat{e}}_{φ} \\ + \frac{1}{r^{2}} (v_{r, θ θ} {\hat{e}}_{r} + v_{r, θ} {\hat{e}}_{θ} + v_{r, θ} {\hat{e}}_{θ} - v_{r} {\hat{e}}_{r} + v_{θ, θ θ} {\hat{e}}_{θ} - v_{θ, θ} {\hat{e}}_{r} - v_{θ, θ} {\hat{e}}_{r} - v_{θ} {\hat{e}}_{θ} + v_{φ, θ θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \sin^{2} θ} (v_{r, φ φ} {\hat{e}}_{r} + \sin θ v_{r, φ} {\hat{e}}_{φ} + \sin θ v_{r, φ} {\hat{e}}_{φ} - \sin^{2} θ v_{r} {\hat{e}}_{r} - \sin θ \cos θ v_{r} {\hat{e}}_{θ} \\ + v_{θ, φ φ} {\hat{e}}_{θ} + \cos θ v_{θ, φ} {\hat{e}}_{φ} + \cos θ v_{θ, φ} {\hat{e}}_{φ} - \sin θ \cos θ v_{θ} {\hat{e}}_{r} - \cos^{2} θ v_{θ} {\hat{e}}_{θ} \\ + v_{φ, φ φ} {\hat{e}}_{φ} - \sin θ v_{φ, φ} {\hat{e}}_{r} - \cos θ v_{φ, φ} {\hat{e}}_{θ} - \sin θ v_{φ, φ} {\hat{e}}_{r} - \sin^{2} θ v_{φ} {\hat{e}}_{φ} \\ - \cos θ v_{φ, φ} {\hat{e}}_{θ} - \cos^{2} θ v_{φ} {\hat{e}}_{φ}) \\ = & (v_{r, r r} + \frac{2}{r} v_{r, r} + \frac{1}{r^{2}} v_{r, θ θ} + \frac{1}{r^{2} \tan θ} v_{r, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{r, φ φ} \\ - \frac{1}{r^{2}} v_{r} - \frac{1}{r^{2}} v_{θ, θ} - \frac{1}{r^{2}} v_{θ, θ} - \frac{1}{r^{2} \tan θ} v_{θ} - \frac{1}{r^{2}} v_{r} - \frac{\cos θ}{r^{2} \sin θ} v_{θ} - \frac{1}{r^{2} \sin θ} v_{φ, φ} - \frac{1}{r^{2} \sin θ} v_{φ, φ}) {\hat{e}}_{r} \\ + (v_{θ, r r} + \frac{2}{r} v_{θ, r} + \frac{1}{r^{2}} v_{θ, θ θ} + \frac{1}{r^{2} \tan θ} v_{θ, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{θ, φ φ} \\ + \frac{2}{r^{2}} v_{r, θ} - \frac{1}{r^{2}} v_{θ} + \frac{1}{r^{2} \tan θ} v_{r} - \frac{\cos θ}{r^{2} \sin θ} v_{r} - \frac{\cos^{2} θ}{r^{2} \sin^{2} θ} v_{θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{φ, φ}) {\hat{e}}_{θ} \\ + (v_{φ, r r} + \frac{2}{r} v_{φ, r} + \frac{1}{r^{2}} v_{φ, θ θ} + \frac{1}{r^{2} \tan θ} v_{φ, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{φ, φ φ} \\ + \frac{2}{r^{2} \sin θ} v_{r, φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{θ, φ} - \frac{\sin^{2} θ + \cos^{2} θ}{r^{2} \sin^{2} θ} v_{φ}) {\hat{e}}_{φ} \\ = & (Δ v_{r} - \frac{2}{r^{2}} v_{r} - \frac{2}{r^{2}} v_{θ, θ} - \frac{2}{r^{2} \tan θ} v_{θ} - \frac{2}{r^{2} \sin θ} v_{φ, φ}) {\hat{e}}_{r} \\ + (Δ v_{θ} + \frac{2}{r^{2}} v_{r, θ} - \frac{1}{r^{2} \sin^{2} θ} v_{θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{φ, φ}) {\hat{e}}_{θ} \\ + (Δ v_{φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{θ, φ} - \frac{1}{r^{2} \sin^{2} θ} v_{φ} + \frac{2}{r^{2} \sin θ} v_{r, φ}) {\hat{e}}_{φ} \end{aligned}$

Native

phab:T375295

$\begin{aligned} (\frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}}) \cdot (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ = & \frac{\partial^{2}}{\partial r^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{2}{r} \frac{\partial}{\partial r} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} \frac{\partial}{\partial θ} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}} (v_{r} {\hat{e}}_{r} + v_{θ} {\hat{e}}_{θ} + v_{φ} {\hat{e}}_{φ}) \\ = & v_{r, r r} {\hat{e}}_{r} + v_{θ, r r} {\hat{e}}_{θ} + v_{φ, r r} {\hat{e}}_{φ} + \frac{2}{r} v_{r, r} {\hat{e}}_{r} + \frac{2}{r} v_{θ, r} {\hat{e}}_{θ} + \frac{2}{r} v_{φ, r} {\hat{e}}_{φ} \\ + \frac{1}{r^{2}} \frac{\partial}{\partial θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial}{\partial φ} (v_{r, φ} {\hat{e}}_{r} + \sin θ v_{r} {\hat{e}}_{φ} + v_{θ, φ} {\hat{e}}_{θ} + \cos θ v_{θ} {\hat{e}}_{φ} + v_{φ, φ} {\hat{e}}_{φ} - \sin θ v_{φ} {\hat{e}}_{r} - \cos θ v_{φ} {\hat{e}}_{θ}) \\ = & v_{r, r r} {\hat{e}}_{r} + v_{θ, r r} {\hat{e}}_{θ} + v_{φ, r r} {\hat{e}}_{φ} + \frac{2}{r} v_{r, r} {\hat{e}}_{r} + \frac{2}{r} v_{θ, r} {\hat{e}}_{θ} + \frac{2}{r} v_{φ, r} {\hat{e}}_{φ} \\ + \frac{1}{r^{2}} (v_{r, θ θ} {\hat{e}}_{r} + v_{r, θ} {\hat{e}}_{θ} + v_{r, θ} {\hat{e}}_{θ} - v_{r} {\hat{e}}_{r} + v_{θ, θ θ} {\hat{e}}_{θ} - v_{θ, θ} {\hat{e}}_{r} - v_{θ, θ} {\hat{e}}_{r} - v_{θ} {\hat{e}}_{θ} + v_{φ, θ θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \tan θ} (v_{r, θ} {\hat{e}}_{r} + v_{r} {\hat{e}}_{θ} + v_{θ, θ} {\hat{e}}_{θ} - v_{θ} {\hat{e}}_{r} + v_{φ, θ} {\hat{e}}_{φ}) \\ + \frac{1}{r^{2} \sin^{2} θ} (v_{r, φ φ} {\hat{e}}_{r} + \sin θ v_{r, φ} {\hat{e}}_{φ} + \sin θ v_{r, φ} {\hat{e}}_{φ} - \sin^{2} θ v_{r} {\hat{e}}_{r} - \sin θ \cos θ v_{r} {\hat{e}}_{θ} \\ + v_{θ, φ φ} {\hat{e}}_{θ} + \cos θ v_{θ, φ} {\hat{e}}_{φ} + \cos θ v_{θ, φ} {\hat{e}}_{φ} - \sin θ \cos θ v_{θ} {\hat{e}}_{r} - \cos^{2} θ v_{θ} {\hat{e}}_{θ} \\ + v_{φ, φ φ} {\hat{e}}_{φ} - \sin θ v_{φ, φ} {\hat{e}}_{r} - \cos θ v_{φ, φ} {\hat{e}}_{θ} - \sin θ v_{φ, φ} {\hat{e}}_{r} - \sin^{2} θ v_{φ} {\hat{e}}_{φ} \\ - \cos θ v_{φ, φ} {\hat{e}}_{θ} - \cos^{2} θ v_{φ} {\hat{e}}_{φ}) \\ = & (v_{r, r r} + \frac{2}{r} v_{r, r} + \frac{1}{r^{2}} v_{r, θ θ} + \frac{1}{r^{2} \tan θ} v_{r, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{r, φ φ} \\ - \frac{1}{r^{2}} v_{r} - \frac{1}{r^{2}} v_{θ, θ} - \frac{1}{r^{2}} v_{θ, θ} - \frac{1}{r^{2} \tan θ} v_{θ} - \frac{1}{r^{2}} v_{r} - \frac{\cos θ}{r^{2} \sin θ} v_{θ} - \frac{1}{r^{2} \sin θ} v_{φ, φ} - \frac{1}{r^{2} \sin θ} v_{φ, φ}) {\hat{e}}_{r} \\ + (v_{θ, r r} + \frac{2}{r} v_{θ, r} + \frac{1}{r^{2}} v_{θ, θ θ} + \frac{1}{r^{2} \tan θ} v_{θ, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{θ, φ φ} \\ + \frac{2}{r^{2}} v_{r, θ} - \frac{1}{r^{2}} v_{θ} + \frac{1}{r^{2} \tan θ} v_{r} - \frac{\cos θ}{r^{2} \sin θ} v_{r} - \frac{\cos^{2} θ}{r^{2} \sin^{2} θ} v_{θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{φ, φ}) {\hat{e}}_{θ} \\ + (v_{φ, r r} + \frac{2}{r} v_{φ, r} + \frac{1}{r^{2}} v_{φ, θ θ} + \frac{1}{r^{2} \tan θ} v_{φ, θ} + \frac{1}{r^{2} \sin^{2} θ} v_{φ, φ φ} \\ + \frac{2}{r^{2} \sin θ} v_{r, φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{θ, φ} - \frac{\sin^{2} θ + \cos^{2} θ}{r^{2} \sin^{2} θ} v_{φ}) {\hat{e}}_{φ} \\ = & (Δ v_{r} - \frac{2}{r^{2}} v_{r} - \frac{2}{r^{2}} v_{θ, θ} - \frac{2}{r^{2} \tan θ} v_{θ} - \frac{2}{r^{2} \sin θ} v_{φ, φ}) {\hat{e}}_{r} \\ + (Δ v_{θ} + \frac{2}{r^{2}} v_{r, θ} - \frac{1}{r^{2} \sin^{2} θ} v_{θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{φ, φ}) {\hat{e}}_{θ} \\ + (Δ v_{φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} v_{θ, φ} - \frac{1}{r^{2} \sin^{2} θ} v_{φ} + \frac{2}{r^{2} \sin θ} v_{r, φ}) {\hat{e}}_{φ} \end{aligned}$

Widetilde & Overline

Widetilde

Serverseitiges Rendering

{\widetilde {\operatorname {div} }}T=\operatorname {\widetilde {div}} T=\operatorname {div} (T^{\top })

Natives MathMl

\tilde{div} T = \tilde{d i v} T = div (T^{⊤})

phab:T352609 und phab:T382674

Overline

Serverseitiges Rendering

{\frac {1}{7}}=0,{\overline {142857}}

Natives MathMl

\frac{1}{7} = 0, \overline{142857}

phab:T352698

Adjungierter Dolbeault-Operator

Serverseitiges Rendering

{\overline {\partial }}^{*}

Natives MathMl

\overset{*}{\overline{\partial}}

phab:T382796

Overarc

0, \overset{⏜}{142857}

Potenzen & Indizes

MathMl

$\sigma _{a}^{2}$
$z=\operatorname {erf} ^{-1}(p)$
$f^{\sharp }$

Native

$σ_{a}^{2}$
$z = \erf^{- 1} (p)$
$f^{♯}$

Partial

Serverseitiges MathMl

$\partial _{x}^{\alpha }$

Natives MathMl

$\partial_{x}^{α}$

Prime

Serverseitiges MathMl

$a'a''f'\mu '$
$a^{\prime }a^{\prime \prime }f^{\prime }\mu ^{\prime }$

Natives MathMl

$a^{'} a^{″} f^{'} μ^{'}$ phab:T385948
$a^{'} a^{''} f^{'} μ^{'}$

Prime nach Determinante

Serverseitiges MathMl

$\det '$
$\det ^{\prime }$

Natives MathMl

${\det^{'}}^{'}$ phab:T386562
$\det^{'}$

Prime nach Operatorname

Serverseitiges MathMl

$\operatorname {determinante} '$
$\operatorname {determinante} ^{\prime }$

Natives MathMl

${determinante}^{'}$
${determinante}^{'}$

Abstände

Minus als Vorzeichen

Serverseitiges MathMl

${\sqrt {-1}}$
$-5$
$-A$
$x-\mathrm {i} y$

Natives MathMl

$\sqrt{- 1}$
$- 5$
$- A$
$x - i y$

Definition

Serverseitiges MathMl

$i:=e_{1}e_{2}$

Natives MathMl

$i : = e_{1} e_{2}$ phab:T382680

Faktor und Funktion

Serverseitiges MathMl

A\exp(x)

Natives MathMl

A \exp (x)

phab:T382642

Klammer und senkrechter Strich

Serverseitiges MathMl

\max_{| z | \leq 1} (| P^{'} (z) |) \leq n \cdot \max_{| z | \leq 1} (| P (z) |)

Natives MathMl

\max_{| z | \leq 1} (| P^{'} (z) |) \leq n \cdot \max_{| z | \leq 1} (| P (z) |)

Kommutative Diagramme

Fehler beim Parsen (Unbekannte Funktion „\begin{tikzcd}“): {\displaystyle \begin{tikzcd} A \arrow[r, "f"] \arrow[d, "g"'] & B \arrow[d, "h"] \\ C \arrow[r, "k"] & D \end{tikzcd} }

Fehler beim Parsen (Unbekannte Funktion „\begin{CD}“): {\displaystyle \begin{CD} A @>a>> B\\ @VVbV @VVcV\\ C @>d>> D \end{CD} }